Verification of Sampling Theorem Validity and Features for Specific Cases of Transformed Samples Number

 
Khanyan G.S. ()
 
Abstract - The work continues the author’s research [7]-[9] on generalization of the time-domain sampling theorem [1], [6] to a finite duration signal in a limited frequency band. The object of the transformation is a harmonic signal s(t) = a cos (2ft+) of limited duration T sampled with frequency F so that its digital realization s(tk) is a sequence of N = FT readings obtained at equidistant time moments tk= t0+k/F; k=0,1,…,N–1, beginning with an arbitrary start time t0. The kernel of interpolating linear transformation (2), along with the number of samples N contains a dimensionless parameter G – the index of frequency band, which, together with the signal frequency f determines all the conditions for the theorem validity and its disturbing effects – the signal frequency and phase shift, the widely known in literature [2]-[5] frequency overlay, and the paradoxical phenomenon described by the author for the first time – complete disappearance (zeroing) of the signal. However, the solutions obtained in the form of “branches” whose numbers I, I, I are not expressed explicitly but are suitable integers that satisfy certain inequalities with respect to the dimensionless frequency f /F (whose boundaries depend on the integer [G] and fractional {G} parts of the index G) cannot be satisfactorily considered as a final stage in developing this direction of interpolation theory. Therefore, the present paper sets a goal, which may seem naive in terms of mathematics but which is quite promising from the point of view of methodology, i.e., to verify the general method for the signal s(tk) transform calculation for specific cases of its discrete samples number N (from 1 to 4), which still allow one to carry out cumbersome calculations. It is in the process of solving this problem that new ideas appear on further development of the Fourier analysis theory with the prospect of developing new applications for digital signal processing in microelectronics. Section II presents the inequalities (4), (6), (7) describing effects listed above – the displacement (3), overlay (5) and signal disappearance s(t) = 0. In fact, only the zeroth branch satisfies the sampling theorem: for I = 0, formula (3) takes the form s(t) = s(t). A technique for direct calculation of sum (2) demonstrated in Section III for specified number of summands N differs radically from the general method described in [9] which is common for any N. The principal difference for specific cases N is the explicit form of the results (13)-(16), (19), (21)-(23) obtained by using well-known trigonometric formulas: cosine of sum, sum of cosines, etc. Such concepts as branch number I, as well as an imposition (overlapping) of branches I and I are absent. The verification carried out in Section IV consists in comparing the sub-cosine expressions of the transformed signal obtained by general and special methods, expressing the branch numbers through the parameters G and f, substituting these expressions in inequalities describing the special effects and proving their validity in each particular case N. As a result of such verification, all the special phenomena (3)-(7) taking place already for the minimum number of samples N=2 are identified. Of particular theoretical and practical interest are formulas (14), (15) and (22), showing that such different phenomena of sampling theorem violation as the frequency overlap and signal disappearance are determined only by the parity of a combined parameter [G]+[fT ] included in the same analytical formula, so that the possibility to generalize it to an arbitrary number of samples N will remove the problem of "branch ambiguity" thus presenting the proof of the theorem in a more "natural" way.

Keywords - sampling theorem, frequency band index, general and private conversion method, branch overlay, signal disappearance.

Верификация справедливости и особенностей теоремы отсчетов для частных случаев числа преобразуемых отсчетов

 
Ханян Г.С. (ФГУП "Центральный институт авиационного моторостроения им. П.И. Баранова", г. Москва)
 
Аннотация - Работа продолжает исследования автора в области теоремы отсчетов для сигнала конечной длительности в ограниченной полосе частот. С возможной полнотой проводится верификация общего метода преобразования сигнала для частных случаев небольшого числа его отсчетов, в ходе выполнения которой определяются дальнейшие направления развития теории.

Ключевые слова - теорема отсчетов, индекс полосы частот, общий и частный метод преобразования, наложение ветвей, исчезновение сигнала.